Phase transitions in economic inequality: Taxation and extremal replacement dynamics

We present a minimal agent-based model of interacting agents characterized by their wealth to study taxation and inequality in a non-conservative economy. Wealth evolves through an extremal stochastic replacement process in which the poorest agent has its wealth replaced by a new random value, financed through a collective taxation mechanism. We explore taxation regimes ranging from regressive to progressive schemes and tune the overall redistribution strength. Under regressive taxation, the system self-organizes into two distinct stationary phases when changing the total tax collected: a non-ergodic, high-inequality regime characterized by wealth condensation in a subset of agents that permanently escape replacement, and a more homogeneous ergodic phase in which all agents participate in the dynamics. Increasing taxes drives an abrupt transition between these phases. The transition is discontinuous and exhibits hysteresis and bistability, consistently detected through the Gini index, the Top $1\%$ wealth share, the entropy, and the Binder cumulant. In contrast, neutral and progressive taxation suppress persistent wealth concentration, preventing the emergence of strongly unequal states and eliminating hysteretic behavior. We complement the simulations with a parameter-free mean-field theory of the wealth dynamics that reproduces the stationary distributions and explains the transition by locating the two spinodals that bound the bistable region. These results show that minimal stochastic redistribution mechanisms alone can produce discontinuous transitions, metastability, and non-ergodicity, demonstrating that taxation structure can determine the emergence and stability of macroscopic inequality.